A shorter stem will move you toward the rear and remove weight from your front wheel. Ths is often good for stability in decending and turns. To avoid the wheel lift when you're climbing you can move forward to the front of your saddle as needed to maintain a good balance point for traction.
Removing spacers or flipping the stem for a down angle moves weight forward. It may work with a shorter stem for a good balance point and stability.

A shorter stem will be better for downhill, but won't solve your problem. It will give you a bit quicker control in corners. Doing a quick google gives this: [Oh dear I don't have enough posts to be allowed to post links, search bing for short stem vs long stem and third link down (at least on my search from the UK) is a youtube video]

I've found that I'm not able to change around my cockpit much if at all. My handlebars have to be even with the seat or I get hand or back pain. The same is true for reach, the exeption being that I can shorten the stem as the bars go longer. If you are comfortable on your bike, I would concentrate on technique (weight forward/seat nose in your ass). Good luck.

My front wheel lifts up during very steep uphill, Do I remove some spacers and try or change it to a shorted stem?

No, you just move your body a bit forward to put more weight on the front wheel.

I'm guessing that you're in the lowest gear available, staying in the saddle, pedaling hard and the pure torque from the low gear makes you pop a wheelie.

One way to counter that is to slide forward so the nose of the saddle is basically in your ass. Sounds crazy, but a position like that moves quite a bit weight forward and deals with the wheelie problem.

Another way to deal with the situation is to stand up and mash your way up the hill in a taller gear. You'll get up the hill a lot faster and wheelies will not be a problem.

Forget about swapping parts on your bike if the bike generally suits you, popping wheelies on a climb is a matter of body position, not whether your stem is a couple of mm longer or shorter.

No, you just move your body a bit forward to put more weight on the front wheel.

I'm guessing that you're in the lowest gear available, staying in the saddle, pedaling hard and the pure torque from the low gear makes you pop a wheelie.

One way to counter that is to slide forward so the nose of the saddle is basically in your ass. Sounds crazy, but a position like that moves quite a bit weight forward and deals with the wheelie problem.

Another way to deal with the situation is to stand up and mash your way up the hill in a taller gear. You'll get up the hill a lot faster and wheelies will not be a problem.

Forget about swapping parts on your bike if the bike generally suits you, popping wheelies on a climb is a matter of body position, not whether your stem is a couple of mm longer or shorter.

When I first started riding 3 years ago I had the same issue on uphills, seemed I could never keep the front wheel on the ground during steep sections. I think Sandrenseren hit the nail on the head with technique likely being the culprit.

For me it felt more natural at first to put more weight on the rear wheel on climbs to maintain traction but you really need to move around to find the "sweet" spot where you still have plenty of rear traction but the front stays down. It also helps to have a smooth pedal stroke, if you're trying too hard to mash on each downstroke you'll be more likely to lift the front even if your weight is balanced well.

^ But Sandrenseren did write "if the bike generally suits you", which to me is the opposite of "bike isn't fitted right".

Another factor besides the front/rear weight shifting is the height of the center of gravity. Basically you duck on the bike. I sometimes do this when standing and pushing the bars down would cause slipping in the rear, and weight in the rear would lead to wheelies. When you're crouched low on the bike, you can retain traction in the rear while keeping the front down as well.

^ But Sandrenseren did write "if the bike generally suits you", which to me is the opposite of "bike isn't fitted right".

Another factor besides the front/rear weight shifting is the height of the center of gravity. Basically you duck on the bike. I sometimes do this when standing and pushing the bars down would cause slipping in the rear, and weight in the rear would lead to wheelies. When you're crouched low on the bike, you can retain traction in the rear while keeping the front down as well.

Yes, I recognized that but there is an issue and it's either technique or bike fit. The OP doesn't seem to have a general understanding of how different size parts effect the over all performance of said bike. This has me thinking they should have a LBS make sure the bike is properly fitted to them. Once that is done, then work on techniques.

We don't even know what type of bike the OP is riding.
Hardtail or full suspension?
If it's FS and your rear suspension isn't set up properly, stem length will be the least of your concerns when climbing.
Stem length has very little to do with how well a bike can be set up for climbing.
Traditional thinking is longer stem = better climbing.
What's a long stem these days?
100mm?
90mm?
70m?
Lots of people are climbing crazy steep and technical trails on 50mm stems or shorter.

I've never liked to slide forward on the saddle to shift my weight forward on my bike. Not that it doesn't work; I just don't like to do it.

My approach is, without moving my hips, to lean forward over the bike more. Basically I just get really low over the handlebars.

Monkeying with the cockpit setup can certainly facilitate either of these techniques. But it makes it harder to lift the front wheel, which is desirable in other circumstances. All life is compromise.

OP, I'd certainly encourage you to try some different cockpit setups. It's good to have an understanding of how this effects the bike, and I think that takes experiencing it. It's also a bit different for each of us.

Instead of "removing" spacers, you can easily swap them from under the stem to above the stem. You actually get a stronger interface of stem to steerer if you have a spacer on top of the stem. If it really bothers you you can always cut the steerer when you find the right position for your stem, but keep in mind that you can't make the steerer longer if you change your mind down the road.

You should set up your cockpit for fit and feel not for climbing position. I also move forward onto the nose of the saddle for extreme climbs or simply just bend my elbows and drop my chest down lower for less extreme climbs.

Don't you hate it when a sentence doesn't end the way you think it octopus?

I had a 26er that I couldn't keep the front wheel down on. Putting a shorter, steeper stem on helped keep the front wheel down. The bike was almost un-rideable for me before (used to 29ers). Much better after the stem change.